Rational points on Atkin-Lehner quotients of Shimura curves

We are interested in proving the triviality of rational points over
Atkin-Lehner's quotients of Shimura curves. In fact it is conjectured
that, except for finitely many exceptions, these quotients only have
special rational points.

Let p, q be prime numbers. We consider the quotient of the Shimura
curve X^{pq}, of discriminant pq,  by the Atkin-Lehner involution w_q.
Under certain explicit congruence conditions, known as the "cas non
ramifié de Ogg", Parent and Yafaev have found a criterion for the
non-existence of non-special rational points over such a quotient. In
their criterion, they relate the problem to some combinatorics
properties of the special fiber of the Cerednick-Drinfeld's model
X^{pq} over Z_p. However, these properties are hard to verify in
particular cases.

We explain how, in a recent work, we manage to verify these properties
in wide generality. We show that the quotient of X^{pq} by w_q has no
non-special rational point for q>245 and p large enough compared to q,
in the "cas non ramifié de Ogg".